Regularization Theory and Shape Constraints

Abstract

Many problems of early vision are ill-posed; to recover unique stable solutions regularization techniques can be used. These techniques lead to meaningful results, provided that solutions belong to suitable compact sets. Often some additional constraints on the shape or the behavior of the possible solutions are available. This note discusses which of these constraints can be embedded in the classic theory of regularization and how, in order to improve the quality of the recovered solution. Connections with mathematical programming techniques are also discussed. As a conclusion, regularization of early vision problems may be improved by the use of some constraints on the shape of the solution (such as monotonicity and upper and lower bounds), when available.